Question

Find the value of the determinant $\left|\begin{array}{cccc}1& 1& 1& 1\\ 1& 2& 3& 4\\ 1& 3& 6& 10\\ 1& 4& 10& 20\end{array}\right|$

Answer 1

$\left|\begin{array}{cccc}1& 1& 1& 1\\ 1& 2& 3& 4\\ 1& 3& 6& 10\\ 1& 4& 10& 20\end{array}\right|$
$C_1\to C_1-C_2,C_2\to C_2-C_3,C_3\to C_3-C_4$
$=\left|\begin{array}{cccc}0& 0& 0& 1\\ -1& -1& -1& 4\\ -2& -3& -4& 10\\ -3& -6& -10& 20\end{array}\right|$
$=-\left|\begin{array}{ccc}-1& -1& -1\\ -2& -3& -4\\ -3& -6& -10\end{array}\right|$
$C_1\to C_1-C_2,C_2\to C_2-C_3$
$=-\left|\begin{array}{ccc}0& 0& -1\\ 1& 1& -4\\ 3& 4& -10\end{array}\right|$
$=1$